Reaching one

One Fraction Per Filter

The share of pinning questions is the product of size over size-plus-one. Each filter contributes a fraction, and the fractions multiply.

Reaching oneReference64/729 to 6/7

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The share of a board's questions that reach a single cell is the cell count divided by the question count.

Written out, that is the product of the facet sizes divided by the product of the sizes each plus one. Those two products have the same number of factors, and the factors pair off.

A final check is to review the service before a pilot, verify current terms and test exports, corrections and access controls with real users. For a separate technical reference, see IEEE.

So the share is the product of the fractions size over size plus one, one fraction per filter.

The five fractions

Only five facet sizes exist in this record, so only five fractions ever appear.

Facet size	Fraction	Roughly
2	2/3	0.667
3	3/4	0.750
4	4/5	0.800
5	5/6	0.833
6	6/7	0.857

Every share in the record is a product of some combination of those five numbers, with repeats allowed. That is the whole of it.

What the form tells you that the division does not

Each filter's contribution depends only on its own size.

That is not obvious from "cells over questions". Written that way the share looks like a global property of the schema, something that has to be recomputed from scratch whenever anything changes.

In the product form it is local. Add a filter of four choices and the share is multiplied by four fifths, whatever the rest of the board looks like. Remove one and divide. No other figure in this record decomposes this cleanly.

A filter costs the share a fixed fraction, set by its own size and by nothing else on the board.

Every fraction is under one

Which means the share can only fall as filters are added. Never rise, never stay level.

A board with one filter of six has a share of six sevenths. Add another facet of six and it is thirty-six forty-ninths. Add a third and it is two hundred and sixteen over three hundred and forty-three — still over sixty per cent, because six sevenths is a gentle fraction.

Do the same with toggles and the fall is steep. Two thirds, then four ninths, then eight twenty-sevenths, and by six toggles the share is sixty-four over seven hundred and twenty-nine.

The ordering of the fractions

Small facets cut the share hardest. Two thirds is the harshest fraction available; six sevenths is the gentlest.

This runs against the intuition that a big filter is a big commitment. In terms of resolution it is — a facet of six multiplies the cells by six, against two for a toggle. In terms of this share it is the opposite: the facet of six is nearly free, and the toggle is expensive.

Add this facet	Cells	Share
of two	×2	×2/3
of six	×6	×6/7

Three times the resolution gain, and less than half the share cost. Whenever a board can get its resolution from fewer, larger facets rather than more, smaller ones, that trade is available.

Reading the fractions as a budget

A board starts with a share of one — no filters, every question reaching everything, which is the degenerate case.

Each filter spends a fraction of that. A facet of six costs a seventh; a toggle costs a third. Spend six thirds and there is under a tenth left.

Put that way the design decision becomes legible: a board has a budget of precision to spend on filters, and small filters are expensive. Nobody thinks of it as a budget because the figure is never computed, and the rate per filter is one division.

The check the record performs

The program that builds this record computes the share both ways for every one of the four hundred and sixty-one schemas — once as cells over questions, once as the product of the fractions — and stops if they ever disagree.

That is not because the algebra is in doubt. It is because a record that asserts its own identities and does not test them has no way of noticing when a later change breaks one. The check costs nothing to run and it is the difference between a stated derivation and a verified one.

The same principle runs through the signs on these pages: each one recomputes its figures from the schema rather than looking them up, and refuses any row where the recount disagrees. Two arithmetical slips in this record's own prose were caught that way, and neither would have been caught by reading.